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# newassign6 - Problem Set 06 Note: The following problem set...

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Problem Set 06 Note: The following problem set is due October 17 by midnight. Please return directly to me in my oFce Rutherford 321. If I’m not there slip your assignment below my door. 1. Let M be the matrix representation of an operator in a basis of a Hilbert space spanned by k -orthonormal vectors. The matrix M is a particular non-diagonal matrix in this space with k -eigenvalues ² k such that ² k << 1. Show that the following determinant identity: p det (1 + M ) = X m =0 1 2 m m ! " tr ± X n =1 M n n ² # m holds no matter what representation of the operator M we want to study. The symbol tr denotes the trace of sum of n powers of the matrix M . 2. Give short answers to the following questions: (a) Consider a Hilbert space spanned by j = 3 2 states. Determine the matrix representation of any generic operator F ( J x ) in this space given that F ( x ) is an analytic function of x . (b) A Hilbert space is spanned by 2

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## This note was uploaded on 12/15/2010 for the course PHYS 357 taught by Professor Keshavdasgupta during the Fall '05 term at McGill.

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newassign6 - Problem Set 06 Note: The following problem set...

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